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Suppose $B_1$ is the unit ball centered at the origin in $ R^N$ with $N \ge 3$. Let $ q= 2^* = \frac{2N}{N-2}$. Does there exist some $C>0$ such that $$\int_{B_1} \frac{ u(x)^2}{|x|^2} dx \le C \| u \|_{L^q(B_1)}^2,$$ for all smooth compactly supported (in $B_1$) radial functions? I assume the answer is false but a counter example maybe isn't completely trivial since both sides scale the same (or maybe it is trivial). If it holds then is $C$ known?

I tried the obvious thing with Hölder's inequality but of course you just miss; maybe one can use weak $L^p$ spaces? any comments greatly appreciated.

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  • $\begingroup$ The function $u(x)=\frac{1}{|x|^{(N/2)-1} |\log |x||^{1/2}}$ is a counterexample in the ball of radius $1/2$ $\endgroup$
    – Giorgio Metafune
    Commented Jul 4 at 16:28
  • $\begingroup$ Thank you very much for the counter example. $\endgroup$
    – Math604
    Commented Jul 8 at 5:01

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You can estimate $\||x|^{-1}u\|_{L^2}\le C \||x|^{-1}\|_{L^{n,\infty}} \|u\|_{L^{q,2}}$ in terms of the Lorentz norm $L^{q,2}$ with $q=\frac{2n}{n-2}$. This is slightly stronger than $L^q$, but not too much. This works also on the ball (or any domain).

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  • $\begingroup$ Thank you very much for your answer. Are you able to control this norm on the right by the Dirichlet energy also or ? Also, i am trying to ''accept'' your answer but i seem to have forgotten how (eventually i will figure it out). $\endgroup$
    – Math604
    Commented Jul 3 at 19:37
  • $\begingroup$ Now i remember how..thanks. $\endgroup$
    – Math604
    Commented Jul 3 at 19:38
  • $\begingroup$ Of course, the $L^{q,2}$ norm is controlled by the $L^2$ norm of $\partial u$. This is a Sobolev embedding with values in Lorentz spaces $\endgroup$
    – Piero D'Ancona
    Commented Jul 3 at 19:40
  • $\begingroup$ Okay, thanks again. I am not familiar at all with these other spaces. $\endgroup$
    – Math604
    Commented Jul 3 at 19:42

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